I developed a theorem (imcomplete) sometime ago, that if you have anintegrallinear combinationofirrationaldistinctradicals equal to each other then there corresponding parts are equal.

For example, if

where are integers then,

.

Now if you have,

where are integers is just impossible,

If you have that,

where are integers then,

Thus,

Then,

is just impossible in integers.

This is my reasoning that why I would not expect a simpler solution with the integers.